Sort of. They're official battles against trainers who are considered masters of a specific type. Upon victory, you're given a badge to prove you won- though most matches are also broadcasted, I've noticed, kind of like a sporting event- and if you collect all the badges of a region, you get to challenge the Elite 4, said to be the best trainers in the land.

Off-worlders who beat the Elite 4 are given the opportunity to make their own gym too- but you need the "Indigo" League badges rather than the "Aurora" League.

[ Unfortunately she hasn't gotten to challenge a single gym yet- things just keep happening and it's awful!! ]